A linear quadratic regulator weight selection algorithm for robust pole assignment

Abstract

Abstract : An advantage of linear quadratic regulator (LQR) design is that it gives a robust system by guaranteeing stability margins. This property is used to develop an algorithm for placing robust poles. The algorithm chooses the positive semidefinite weighting matrix Q and positive definite weighting matrix R by attempting to place closed loop poles at a set of desired poles. If the desired poles lie outside the allowable LQR region, the algorithm finds the achievable poles inside the region that are closest to the desired poles. The solution requires using a gradient search technique to minimize a weighted eigenvalue difference cost function. The weighting of the eigenvalue difference establishes the relative importance between the poles. In a multi-input multi- output system, the placement of one pole effects the allowable placement region of the other poles. Thus, the heavier weighted poles have precedence and are forces closer to their desired location. The algorithm is programmed to run on the software package MATLAB and the related subroutines are discussed. Several examples are included to illustrate the use of the algorithm, some of which can be solved in closed form to compare with the algorithm's solution. The results show that this technique is accurate for selecting robust poles at or close to he desired pole locations.